Josh Buli
MATH 008A - Lecture Notes 11
2014
These notes cover the lecture on § 6.4 - Sine and Cosine Functions (specifically cosine), and the graphs in
§ 6.5 - Tangent, Cotangent, Cosecant, and Secant. These are some guidelines to the graphs if you need to
sketch them. Examples are given for the cosine graphs, as being able to sketch these using transformations
will be required knowledge.
1. § 6.4 - Sine and Cosine Functions
Definition 1.1. We can define the cosine function as a horizontal shift of sin(x), since we know the graph
of this function already:
π
f (x) = cos(x) = sin(x + )
2
where the horizontal shift is π2 units to the left. The graph is given in the plot below. Note the zeros are at
3π
π
2 and 2 .
Figure 1. Graph of f (x) = cos(x)
Definition 1.2. In general, we can write a trig function, say sine, as
y(t) = A sin(ωt − c)
where A = amplitude, ω = frequency, and
T = 2π
ω for sin and cos.
c
ω
= φ = phase shift. The period, T is given by the formula
Example 1. Sketch a complete graph of f (x) = −4 cos(2x) − 2
Solution: We want to do this graph by the graph transformations that are given in a previous lecture. First
we calculate 3 quantities: Amplitude, Period, and vertical shift. Recall that the function has the general
2π
form: f (x) = A cos(ωx) + b. So then we have that |A| = 4 is the amplitude, P = 2π
ω = 2 = π, and
b = −2. From this, we can deduce the graph. Starting out with the parent graph above in figure 1, we apply
transformations as follows:
1
2
Figure 2. Graph Transformations: (1) f (x) = cos(x), (2) f (x) = cos(2x), (3) f (x) =
− cos(2x), (4) f (x) = −4 cos(2x), (5) f (x) = −4 cos(2x) − 2. From left to right and top to
bottom.
3
Example 2. Sketch a complete period: y = 4 sin (4x) + 1
Example 3. Sketch a complete period: y = 2 cos (6x) − 3
Example 4. Sketch a complete period: y = 3 sin (2x) + 1
Definition 1.3. We can define the tangent function using sin(x) and cos(x), since we know the function
values of these trig functions already:
sin(x)
f (x) = tan(x) =
cos(x)
So we will have vertical asymptotes where cos(x) = 0 and zeros where sin(x) = 0. The graph is shown below
for the zoomed in version and the periodic zoomed out version.
Figure 3. Graph of f (x) = tan(x) zoomed in on left. Graph of f (x) = tan(x) zoomed out
on right.
Definition 1.4. We can define the cotangent function using sin(x) and cos(x), since we know the function
values of these trig functions already:
1
cos(x)
f (x) = cot(x) =
=
tan(x)
sin(x)
So we will have vertical asymptotes where sin(x) = 0 and zeros where cos(x) = 0. The graph is shown below
for the periodic zoomed out version.
Figure 4. Graph of f (x) = cot(x) zoomed out.
4
Definition 1.5. We can define the cosecant function using sin(x), since we know the function values of
this trig functions already:
1
f (x) = csc(x) =
sin(x)
So we will have vertical asymptotes where sin(x) = 0. The graph is shown below for the periodic zoomed out
version.
Figure 5. Graph of f (x) = csc(x) in blue. Graph of f (x) = sin(x) in red.
Definition 1.6. We can define the secant function using cos(x), since we know the function values of this
trig functions already:
1
f (x) = sec(x) =
cos(x)
So we will have vertical asymptotes where cos(x) = 0. The graph is shown below for the periodic zoomed out
version.
Figure 6. Graph of f (x) = sec(x) in blue. Graph of f (x) = cos(x) in red.